
As above
\(\sqrt{ \frac{Nl}{b} } \frac{U_F(b/l) -\vartheta _{N-\frac{Nl}{b}, N} }{\frac{b}{l}U^{\prime }_F(\frac{b}{l})} = O_p(1).\) 

By corollary 1.1.10 in [1]} under assumption  REF
\(\lim _{t\rightarrow \infty } \dfrac{U_F(tx)-U_F(t)}{tU_F^{\prime }(t)} = \log (x),\) 
that is, \(NU_F^{\prime }(N)\)  are the canonical scaling constants \(\alpha _N\)  that appear in eq. (REF ). Hence by lemma  REF
\(\frac{ \frac{b}{l}U^{\prime }_F\left(\frac{b}{l}\right) }{ \vartheta _{b-q, b, T}^{(s)} - \vartheta _{b, b, T}^{(s)}} = O(1)\) 

The last term converges to zero.

